Local Enumeration and Majority Lower Bounds

TL;DR

This work links local enumeration methods for $k$-CNFs to fundamental questions in circuit complexity and satisfiability, introducing Enum($k,t$) as a framework whose upper bounds imply $ ext{Σ}^k_3$-circuit lower bounds and faster $k$-SAT algorithms via run-time bounds such as $O(\, extstyleigl(rac{n}{2}igr) igr)$ and $inom{n}{n/2}/b(n,k,n/2)$.The authors develop a transversal-tree framework and the TreeSearch algorithm, leveraging canonical clause orderings and random edge orderings to enumerate minimum-size transversals with controlled pruning, connecting to hypergraph transversals and Turán-type questions.For monotone $3$-CNFs, they obtain nontrivial survival-bound analyses giving running times like $1.164^n imes 1.9023^t$, and demonstrate a concrete improvement by achieving Enum($3, frac{n}{2}$) in $1.598^n$, which yields a new lower bound $ ext{Σ}^3_3( ext{Maj}) ilde{ ightarrow} 1.251^{n}$.Extending to general $3$-CNFs via fullness and double-marking, they preserve the framework to obtain analogous bounds, and they further apply TreeSearch to CNFs with bounded negations to achieve $1.8204^n$-time enumeration, highlighting a promising path toward stronger lower bounds and faster algorithms.

Abstract

Depth-3 circuit lower bounds and $k$-SAT algorithms are intimately related; the state-of-the-art $Σ^k_3$-circuit lower bound and the $k$-SAT algorithm are based on the same combinatorial theorem. In this paper we define a problem which reveals new interactions between the two. Define Enum($k$, $t$) problem as: given an $n$-variable $k$-CNF and an initial assignment $α$, output all satisfying assignments at Hamming distance $t$ from $α$, assuming that there are no satisfying assignments of Hamming distance less than $t$ from $α$. Observe that: an upper bound $b(n, k, t)$ on the complexity of Enum($k$, $t$) implies: - Depth-3 circuits: Any $Σ^k_3$ circuit computing the Majority function has size at least $\binom{n}{\frac{n}{2}}/b(n, k, \frac{n}{2})$. - $k$-SAT: There exists an algorithm solving $k$-SAT in time $O(\sum_{t = 1}^{n/2}b(n, k, t))$. A simple construction shows that $b(n, k, \frac{n}{2}) \ge 2^{(1 - O(\log(k)/k))n}$. Thus, matching upper bounds would imply a $Σ^k_3$-circuit lower bound of $2^{Ω(\log(k)n/k)}$ and a $k$-SAT upper bound of $2^{(1 - Ω(\log(k)/k))n}$. The former yields an unrestricted depth-3 lower bound of $2^{ω(\sqrt{n})}$ solving a long standing open problem, and the latter breaks the Super Strong Exponential Time Hypothesis. In this paper, we propose a randomized algorithm for Enum($k$, $t$) and introduce new ideas to analyze it. We demonstrate the power of our ideas by considering the first non-trivial instance of the problem, i.e., Enum($3$, $\frac{n}{2}$). We show that the expected running time of our algorithm is $1.598^n$, substantially improving on the trivial bound of $3^{n/2} \simeq 1.732^n$. This already improves $Σ^3_3$ lower bounds for Majority function to $1.251^n$. The previous bound was $1.154^n$ which follows from the work of Håstad, Jukna, and Pudlák (Comput. Complex.'95).

Theorems & Definitions (68)

• Proposition 1
• proof
• Theorem 1: Main result
• Corollary 1
• Theorem 2
• Definition 1: Transversals
• Definition 2: Transversal number
• Proposition 2
• proof
• Definition 3: $(k,X)$-trees